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Addition of fractions is one of the important mathematical operations of fractions.
To add together two fractions, first we need to find equivalent fractions that share a common denominator, then the sum is given by adding the numerators. There are two kinds of addition of fractions:

• Add Fractions with Like Denominators
• Add Fractions with Unlike Denominator

The following rules are helping the student for adding the fractions.

Rule 1 : The least common multiple is determined if the denominator values of fractions are differing.

Rule 2 : The least common multiple is multiplied with numerator value and add those values.

Rule 3 : If we adding the fractions with mixed numbers means first change the mixed number into improper fraction.

## Adding Fractions with Like Denominators

Step 1: Make sure the bottom numbers (the denominators) are the same

Step 2: Add the top numbers (the numerators). Put the answer over the the same denominator.

Step 3: The denominators will remain the denominator of the built up fraction.

Step 4: Simplify or reduce the fraction (if needed)
Here is a example of adding fractions with same denominators

### Solved Example

Question: Add $\frac{1}{4}$ and $\frac{1}{4}$
Solution:
Step 1: Since the bottom numbers (denominator) are same, we need to add the top numbers (numerator).

Step 2: Put the answer over the same bottom number:

$\frac{(1+1)}{4}$ = $\frac{2}{4}$

Step 3: Simplify the fraction: $\frac{2}{4}$

Correct answer is $\frac{1}{2}$

## Adding Fractions with Unlike Denominators

Step 1 : Find the Least Common Denominator (LCD) of the fractions

Step 2 : Rename the fractions to have the LCD

Step 3 : Add the numerators of the fractions

Step 4 : Simplify the Fraction (if needed)
Here is a example of adding fractions with different denominators

### Solved Example

Question: Add the fraction $\frac{3}{8}$ and $\frac{5}{12}$
Solution:
Step1: The least common denominator(LCD) of 8 and 12 = 24

Step 2: First fraction: $\frac{3}{8}$ x $\frac{3}{3}$ = $\frac{9}{24}$

Second fraction: $\frac{5}{12}$ x $\frac{2}{2}$ = $\frac{10}{24}$

Step3: Addition: $\frac{9}{24}$+ $\frac{10}{24}$

= $\frac{(9 + 10)}{24}$

Correct answer is $\frac{19}{24}$

Adding fractions with variables is a little bit difficult, a variable in math is an unidentified number, more often than not represented by a letter such as X, Y, A or B. By using a variable in an equation can help us to find for a missing number.

### Solved Examples

Question 1: To find the fraction with variable of $\frac{2x}{3}$ + $\frac{3x}{3}$
Solution:
Here the two fractions have same denominator, we can easily add the variables in numerator without changing the values of denominator.
= $\frac{2x}{3}$ + $\frac{3x}{3}$
= $\frac{2x + 3x}{3}$
= $\frac{5x}{3}$
So the final answer is $\frac{5x}{3}$

Question 2: To find the fraction with variable of $\frac{2x}{8}$ + $\frac{9x}{8}$
Solution:
Here the two fractions have same denominator, we can easily add the variables in numerator without changing the values of denominator.
$\frac{2x}{8}$ + $\frac{9x}{8}$
= $\frac{2x + 9x}{8}$
= $\frac{11x}{8}$

So the final answer is $\frac{11x}{8}$

## How to Add Fractions with Whole Numbers

Fractions are termed as the ratio between two numbers as $\frac{a}{b}$ . Whole numbers are the numbers without any fraction terms or decimal points. The fraction numbers can be added to the whole numbers by following some simple math methods and can achieve the correct result, which will be the fraction values mostly. Some examples for adding fractions with whole numbers are discussed as below

### Solved Examples

Question 1: Add the fraction value $\frac{1}{3}$ with the whole number 5.
Solution:
The steps to be followed to add the fraction value $\frac{1}{3}$ with the whole number 5 are as follows:
= $\frac{1}{3}$ + 5

= $\frac{1}{3}$ + 5 $\frac{3}{3}$ (multiply and divide the number 3 with the given whole number 5 to convert it into fraction value).

= $\frac{1}{3}$ + $\frac{15}{3}$

= $\frac{1 + 15}{3}$ (take the number 3 as the common divisor for both)

= $\frac{16}{3}$ (it is also a fraction value)

Question 2: Add the fraction value $\frac{5}{4}$ with the whole number 3.
Solution:
The steps to be followed to add the fraction value $\frac{5}{4}$ with the whole number 3 are as follows:
= $\frac{5}{4}$ + 3

= $\frac{5}{4}$ + 3$\frac{4}{4}$ (multiply and divide the number 4 with the given whole number 3 to convert it into fraction value).

= $\frac{5}{4}$ + $\frac{12}{4}$

= $\frac{5 + 12}{4}$ (take the number 4 as the common divisor for both)

= $\frac{17}{4}$ (it is also a fraction value)

Adding Fractions word problems taken place in many real world situations. Below you could see the word problems on adding fractions

### Solved Examples

Question 1: John finished $\frac{1}{4}$ of the work in 2 hours and $\frac{3}{2}$ of the work in 3 hours. How many works has John finished?
Solution:
To find the total work we have to add the two fractions,
Total work = $\frac{1}{4}$ + $\frac{3}{2}$

Here the denominators are different. So we have tot take LCM. LCM of 2, 4 = 4

$\frac{1}{4}$ + $\frac{3}{2}$ = $\frac{1}{4}$ + $\frac{3\times2}{2\times2}$

= $\frac{1}{4}$ + $\frac{6}{4}$

= $\frac{1+6}{4}$

= $\frac{7}{4}$

Hence John has finished $\frac{7}{4}$ of the work.

Question 2: Daisy bought $\frac{5}{2}$ pounds of apple and $\frac{3}{2}$ pounds of cherries. How much fruit did she buy in all?
Solution:
Pounds of apple = $\frac{5}{2}$
Pounds of cherries = $\frac{3}{2}$

To find the total pounds of fruits, we need to add the two fractions.

Total amount of fruits = $\frac{5}{2}$ + $\frac{3}{2}$

= $\frac{5+3}{2}$

= $\frac{8}{2}$ = 4

Hence Daisy bought 4 pounds of fruit in all.